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Calculus
A Rigorous First Course
By Daniel J. Velleman Dover Publications, Inc.
Copyright © 2016 Daniel J. Velleman
All rights reserved.
ISBN: 978-0-486-81885-6
CHAPTER 1
Preliminaries
1.1 Numbers and Sets
The numbers we will use in this book are the real numbers. These are all the numbers that can be written in decimal notation. We often think of them as corresponding to points on a number line (see Figure 1.1). The simplest real numbers are the integers: the numbers 0, 1, -1, 2, -2, 3, -3, and so on. A real number is said to be rational if it can be written as an integer divided by an integer; for example, 2/3 and -13/5 are rational numbers. Notice that every integer is also a rational number, since, for example, we can write 3 as 3/1. Real numbers that are not rational are called irrational. For example, it can be shown that [square root of 2] = 1.41421 ... and π = 3.14159 ... are irrational. Both rational and irrational numbers are spread throughout the number line; in fact, between any two real numbers there are infinitely many rational numbers and also infinitely many irrational numbers.
We will also often work with collections of numbers. In mathematics, a collection of objects is called a set, and the objects in the collection are called elements of the set. The simplest way to specify a set is to list the elements of the set between braces. For example, {-1, 0, 3/2} is the set whose elements are the three numbers -1, 0, and 3/2. If we let the letter A stand for this set, then we write 3/2 [member of] A to say that 3/2 is an element of A, while 4 centsA means that 4 is not an element of A.
Another way to specify a set is to give a rule for determining which objects belong to the set and which do not. For example, if we write
B = {x: 2x3 - x2 - 3x = 0}, (1.1)
then this means that B is the set whose elements are all values of x that satisfy the equation 2x3 - x2 -3x = 0. Equation (1.1) is read "B is equal to the set of all x such that 2x3 - x2 -3x = 0." The equation 2x3 - x2 -3x = 0 that appears in the definition of B is a statement that is true for some values of x and false for others. You should think of it as an elementhood test for the set B; those values of x that make the equation true pass the test and are elements of B, while those that make the equation false are not. To determine which numbers belong to B we simply have to solve the equation, which we can do by factoring the left-hand side. We have
2x3 - x2 -3x = x(x + 1)(2x -3),
so the equation can be rewritten x(x + 1)(2x -3) = 0, and the solutions are 0, -1, and 3/2. These are the elements of the set B. Notice that these are exactly the same as the elements of the set A defined earlier. Thus B = A; they are both the same collection of numbers, described in two different ways.
Although we will usually use the letter x when writing an elementhood test to define a set, as we did in the definition of B, in fact any letter can be used. For example, the set C = {y: 2y3 - y2 -3y = 0} is the set of all values of y that satisfy the equation 2y3 - y2 -3y = 0. Of course, this is the same equation that we used in the definition of B, but with x replaced by y. The values of y that satisfy the equation are therefore once again the numbers 0, -1, and 3/2. Therefore C = B = A; we have the same set of numbers described in yet another way.
Here is another example of a set defined by an elementhood test: I = {x: 2< x< 5}. This time the elementhood test is 2 < x< 5, which is a shorthand way of saying that 2 < x and x< 5. In this case 3 [member of] I, since the statement 2 < 3 < 5 is true, but 5 [not member of]I, since the statement 2 < 5 < 5 is false. The elements of I are all the numbers strictly between 2 and 5. There are infinitely many numbers in this range, so we cannot list all the elements of I, as we did for A. But we can mark them on a number line, as in Figure 1.2.
The set I is an example of a kind of set called an open interval. For any numbers a and b with a < b, the set of all numbers strictly between a and b is an open interval, and it is denoted (a, b). In other words,
(a, b) = {x: a < x < b}.
The numbers a and b are called the endpoints of the interval. Thus I = (2, 5); it is the open interval with endpoints 2 and 5.
The endpoints of an open interval are not elements of the interval. But sometimes we will want to include the endpoints, so we define
[a, b] = {x: a ≤ x ≤ b}.
This set is called a closed interval. For example, [2, 5] = {x: 2 ≤ x ≤ 5}. This set is exactly the same as the open interval (2, 5) considered earlier, except that it includes the endpoints 2 and 5 as elements. If we include only one endpoint, we get a half-open interval. As you might guess, we write half-open intervals like this:
(a, b] = {x: a < x ≤ b}, [a, b) = {x: a ≤ x < b}.
In general, we use a square bracket to indicate that an endpoint is included in an interval, and a parenthesis to indicate that it is not. Figure 1.3 shows examples of closed and half-open intervals.
The interior of an interval is the set containing all numbers in the interval except the endpoints. Thus, the interior of the closed interval [2, 5] is the open interval (2, 5). The interiors of (2, 5] and [2, 5) are also (2, 5), and we will even say that the interior of (2, 5) is (2, 5).
Finally, we sometimes want to consider intervals that extend infinitely far in some direction, so we introduce the notation:
[MATHEMATICAL EXPRESSION OMITTED]
Some examples of infinite intervals are shown in Figure 1.4. The interior of [a, ∞) is (a, ∞), and the interior of (-∞, b] is (-∞, b). The set of all real numbers is often denoted R, but we could also think of it as the interval (-∞, ∞). We consider any interval that does not include its endpoints to be an open interval. Thus, the intervals (a, ∞), (-∞, b), and (-∞, ∞) are open intervals, and the interior of any interval is an open interval.
Since this is our first use of the infinity symbol∞, it might be worthwhile to pause at this point to explain what this symbol means. The most important thing to understand about the infinity symbol is that there is no such number as infinity. You might wonder, then, how it can be correct to use this symbol in mathematical notation like (a,8). The answer is that, according to the definition we have given, this notation is simply a shorthand for {x:x > a}, and this last expression makes no mention of infinity. Every time we make a statement using the symbol ∞, it will be a similar shorthand for a statement that makes no mention of infinity. We will never use the symbol ∞ as if it stood for a number. Thus, for example, we would never set x equal to ∞in some formula, and we would never talk about the "closed interval" [2, ∞].
There are two ways of combining sets that we will sometimes make use of. If A and B are sets, then the intersection of A and B, denoted A [intersection] B, is the set whose elements are those objects that belong to both A and B. Thus
A [intersection] B = {x: x [member of] A and x [member of] B}.
For example,
[MATHEMATICAL EXPRESSION OMITTED]
Looking at Figure 1.4, you can see that the elements of [2, ∞) [intersection](-∞, 5) are those numbers that are in the overlap of the sets [2, ∞) and (-∞, 5). In general, you can think of A [intersection] B as the overlap of A and B.
The union of A and B, denoted A [union] B, is the set whose elements are all those objects that are elements of either A or B (or both). That is,
A [union] B = {x: x [member of] A or x [member of] B}.
You could think of A [union] B as the set you get if you take all the elements of A, and all the elements of B, and throw them together into one set. For example, if we take all the numbers in the intervals (2, 4] and [4, 5] and put them together into one set, we get the interval (2, 5]. That is,
[MATHEMATICAL EXPRESSION OMITTED]
If A and B are sets, then A is called a subset of B if every element of A is also an element of B. We write A [subset or equal to] B to indicate that A is a subset of B. For example, (2, 4) [subset or equal to] [2, 4], and [2, 4] [subset or equal to] (1, 5).
One reason that intervals are important in calculus is that they often come up as solution sets of inequalities. In particular, we will often be concerned with inequalities involving absolute values. Recall that the absolute value of a number x is defined as follows:
[MATHEMATICAL EXPRESSION OMITTED]
This notation means that if x ≥ 0 then |x| = x, and if x< 0 then |x| = -x. For example, |3| = 3 and |-4| = -(-4) = 4.
The fact that |x| is defined by cases, with one formula when x ≥ 0 and a different formula when x< 0, suggests a method that can be used when solving any problem involving absolute values: reasoning by cases. As a simple example of this kind of reasoning, notice that if x ≥ 0 then we have |x| = x ≥ 0, and if x< 0 then |x| = -x > 0. In both cases the statement |x| ≥ 0 is true, so we conclude that for every number x, |x| ≥ 0. You can also use reasoning by cases to show that for every number x, [square root of x2] = |x| (see Exercise 15).
Here's an example of how reasoning by cases can be used to solve an inequality involving absolute values:
Example 1.1.1. Solve:
|x| < 3.
Solution. Motivated by the definition of |x|, we will consider x ≥ 0 and x< 0 separately.
Case 1.x ≥ 0. In this case, according to the definition of absolute value we have |x| = x, so the inequality |x| < 3 means x< 3.
Case 2.x< 0. Now the definition of absolute value says that |x| = -x, and substituting this into our inequality |x| < 3 gives us -x< 3. Multiplying by -1 (and remembering that when multiplying an inequality by a negative number, we must reverse the direction of the inequality) we get x > -3.
So what's the solution to our inequality? Is it x < 3, as we found in case 1, or x > -3, as in case 2? To answer this question, we must think about what it means to solve an inequality. To solve the inequality |x| < 3 means to determine which values of x make the inequality true. Our reasoning in case 1 shows that, for x ≥ 0, the inequality means x< 3. Thus the inequality is true if 0 ≤ x< 3 and false if x ≥ 3. We can't tell from this reasoning whether the inequality is true or false when x< 0; that's the purpose of case 2. Case 2 tells us that, for x< 0, the inequality will be true precisely when x > -3. Thus, the inequality is true if -3 < x< 0 and false if x ≤ -3. Putting all this information together, we conclude that the inequality is true if -3 < x< 3 and false if either x ≥ 3 or x ≤ -3, as shown in Figure 1.5. This means that the solution set of the inequality is an open interval:
{x: |x| < 3} = {x: -3 < x < 3} = (-3, 3).
Notice that in case 1 we determined that all numbers in the interval [0, 3) are in the solution set, and in case 2 we determined that the numbers in (-3, 0) are also in the solution set. The solution set is therefore the union of these two intervals: [0, 3) [union] (-3, 0) = (-3, 3).
Here's another way of describing the answer to Example 1.1.1. Our reasoning shows that the statements |x| < 3 and -3 < x< 3 are true for exactly the same values of x; the two statements are equivalent. In other words, for any number x, if |x| < 3, then -3 < x< 3, and if -3 < x< 3, then |x| < 3. Mathematicians usually describe this situation by saying that |x| < 3 is true if and only if -3 < x< 3. The phrase "if and only if" comes up often in mathematics, and we will see it many times later in this book.
Of course, there is nothing special about the number 3 in this example. Similar reasoning, using the variable y in place of the number 3, can be used to establish the following theorem. Parts 3 and 4 of the theorem follow directly from parts 1 and 2, by negating the statements involved.
Theorem 1.1.2. For any numbers x and y, the following statements are true:
1. |x| < y if and only if -y < x < y.
2. |x| = y if and only if -y ≤ x ≤ y.
3. |x| = y if and only if either x ≤ -y or x ≥ y.
4. |x| > y if and only if either x < -y or x > y.
The most important use of absolute values in calculus is to compute distances between numbers on the number line. To find the distance between two numbers, we subtract the smaller number from the larger. For example, the distance from -4 to 3 on the number line is 3-(-4) = 7. In general, if a ≤ b then the distance from a to b is b-a. But if a > b then the distance is a -b. Is there a single formula that gives the distance from a to b no matter which of the numbers is larger?
It turns out that the formula |b-a| does the trick. We can see this by once again using reasoning by cases.
Case 1.a ≤ b. Then b-a ≥ 0, so |b-a| = b-a, which, in this case, is the distance from a to b on the number line.
Case 2.a > b. Now b-a< 0, so |b-a| = -(b-a) = a -b, which is, once again, the distance from a to b in this case.
Thus, no matter which of the numbers a and b is larger, we have
|b-a| = the distance from a to b on the number line.
This fact provides a nice way to see why our solution to Example 1.1.1 makes sense. We have
|x| = |x -0| = distance from 0 to x on the number line.
With this interpretation for |x|, the inequality |x| < 3 can be thought of as saying
(distance from 0 to x on the number line) < 3.
It is clear geometrically that the values of x for which this statement is true are the numbers in the open interval (-3, 3), exactly as we found in our solution to Example 1.1.1.
Example 1.1.3. Solve:
|3t + 2| ≤ 4.
Solution. By part 2 of Theorem 1.1.2, the inequality to be solved is equivalent to
-4 ≤ 3t + 2 ≤ 4.
Subtracting 2 all the way through the inequality gives us
-6 ≤ 3t ≤ 2,
and then dividing by 3 we get
-2 ≤ t ≤ 2/3.
Thus the solution set for this inequality is a closed interval:
[MATHEMATICAL EXPRESSION OMITTED]
Sometimes it is useful to be able to simplify absolute values of complicated expressions. In such situations, the following theorem can be helpful.
(Continues...)
Excerpted from Calculus by Daniel J. Velleman. Copyright © 2016 Daniel J. Velleman. Excerpted by permission of Dover Publications, Inc..
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